The 23-year-old Queenslander was crowned the new number one on Monday a day after beating Germany's Julia Goerges 6-3 7-5 in the final of the Birmingham Classic on Sunday.
The achievement has capped a remarkable run for Barty, who claimed her first Grand Slam title at Roland Garros two weeks ago.
Goolagong Cawley scaled the peak of the world rankings in 1976 although it only came to light 31 years later after the discovery of an error in the records.
Like Goolagong, Barty is proud of her indigenous Australian heritage, with her father having Ngarigo ancestry through one of his grandmothers.
"I'm nowhere near her status," Barty, said of the seven-times Grand Slam champion. "To be mentioned in the same sentence is incredible.
"Evonne, she's an amazing human being and has set the tone for so many Australians and so many indigenous Australians around our country and around the world.
"She is an amazing person. And what she has done in her career was incredible and what she continues to do off the court for us as a sport is amazing."
Goolagong Cawley was quick to congratulate Barty, whose achievement comes three years after she returned to the tour in 2016 following an indefinite break from the game.
"Ash is a very worthy number one and winning at the French will have given her even more confidence," Goolagong Cawley said in a statement on Monday.
"I am so proud that another Aboriginal player sits on top of the rankings in women's tennis, particularly a young lady who conveys such happiness in all she does.
"She really enjoys being out there and she has become an outstanding and inspirational example to all Australians. Well done Ash - long may your success continue."
Barty's number one status triggered a flood of other tributes in mainstream and social media for a player who transformed herself into a professional cricketer during her time away from tennis.
"A big Congratulations to @ashbar96," American 18-times Grand Slam champion Chris Evert wrote on Twitter.
"Tremendous story, talent, and most importantly, person."
Australian Hollywood actor Russell Crowe tweeted: "Ash Barty. #1. Brilliant."
American women's great Billie Jean King also praised the crafty Australian all-court player.
"With incredible versatility, perseverance, and focus, Ash Barty is an inspiration to the next generation of young players in Australia. Well done!"
Barty will head to Eastbourne on Monday for a last warmup before her bid to claim a maiden Wimbledon title.
(Reporting by Ian Ransom; Editing by Sudipto Ganguly)
